Hydrogen as a Consistency Test: Phase Completion, the Second-Order Clock Term, the Size of the Compton-Scale Structure, and Hyperfine Structure

The electromagnetic metric of Volume I places the potential, through Σ_EM = qΦ/mc² with Φ the electric potential, in three parts of the metric: g₀₀, g_ij = (1 + χΣ_EM)δ_ij and g₀ᵢ. Volume III derives this metric, with every coefficient fixed, from general relativity and the U(1)-coupled phase. Hydrogen tests all three parts, using the metric and the identity of Volume II, h = E Δt: one completed phase cycle carries action h, so a clock of energy E completes a cycle in Δt = h/E. Time part. At order α⁴ the first-order metric leaves the Σ_EM² coefficient of g₀₀ open; the naive choice fails by a factor of eight. Two independent routes fix it: the exact covariant response metric of Volume I, which Volume III derives, and h = E Δt with E including the potential energy. Both give g₀₀ = −(1 + Σ_EM)² and the correct fine structure. Spatial part. The coefficient χ = 1, fixed independently in Volume I and forced in Volume III, removes a cross term between the potential and the kinetic energy; gravity's value in this convention, χ = −2, would shift the ground state by −6.5 meV, thirty-six times the whole fine-structure shift. Vector part. With coefficient 1 in g₀ᵢ (gravity: 4), circulating charge gives the current-loop contact field +⅔μ₀μ, and the ground-state hyperfine splitting comes out within 0.11% for hydrogen (proton moment as input) and 0.21% for muonium (no input beyond the muon mass). Coefficient 2 or 4, a pole dipole, or g = 1 fail by a factor of two or more. With spin–orbit and the Darwin term the Dirac levels follow; g = 2 is Volume III's value for a spinor that carries the charge phase of the Compton cycle, and the Thomas factor is kinematic. The Darwin term is the size of a sphere: the Kerr parameter S/mc quantized for spin ½ has fixed magnitude (√3/2)r₀, an empty centre and equal spread on every axis, and the classical ring it replaces gives one third of the term. Beyond α⁴ the velocity-independent metric is the slow-motion form of Volume I's exact covariant response metric, whose length is the additive (Randers) coupling, exact at every order. In the Lamb shift the large logarithm, with its cutoff set by the Compton cycle, and, given the anomaly, both anomalous-moment pieces match QED, together 96% of the measured shift; the remaining constant and vacuum polarization are taken from QED. These show that the forced metric has the correct structure.